John Kutsko, executive director of the Society of Biblical Literature, said he was worried the Christian organisation was limiting freedom of expression on the internet by determining who can use the domain and who cannot.
"The internet is public space," he said to Religion News Service.
"It's our understanding that .bible was registered to be public space and not have the kind of restrictions that you would expect of a domain that was proprietary or brand-oriented."
Competing with domains ending with ".com", ".org" and others, the ABS has been able to attract at least 1,190 customers to acquire the .bible domain.
Religion News Service reported that ABS had recently enacted a policy prohibiting customers wanting to use the .bible domain from posting any material that "espouses or promotes a religious, secular, or other worldview that is antithetical to New Testament principles, including but not limited to the promotion of a non-Christian religion or set of religious beliefs."
The ABS has been managing the domain since 2016.
A short statement issued by the ABS said it had met with "complaining parties" but said that its policies are "in complete compliance" with the Internet Corporation for Assigned Names and Numbers (ICANN), which manages internet resources and coordinates its domain name system.